Algebraic and model-theoretic properties of tilings

Abstract

We investigate the relations between the geometric properties of tilings and the algebraic and model-theoretic properties of associated relational structures. Isomorphism and local isomorphism of tilings up to translation correspond to isomorphism and elementary equivalence of relational structures. In particular, two Penrose tilings, or two Robinson tilings, are elementarily equivalent. Classical results concerning the local isomorphism property and the “extraction preorder” for tilings are generalized to uniformly locally finite relational structures. Then, we define “equational structures”, which generalize both Cayley graphs of groups and relational structures associated to tilings, and for which we have an appropriate notion of free structure relative to a system of equations. For each finite system Σ of prototiles and local configurations, we give a finite set of local conditions characterizing the connected relational structures which are homomorphic images of Σ-tilings. It follows that tilings are free relative to finite systems of equations which express these conditions. We also prove that the theory of a tiling is superstable, model-complete, and can be axiomatized by ∀∃ sentences. One ∀∃ sentence suffices in the case of Penrose tilings or Robinson tilings.